Congressional Apportionment: A Liberal Arts Perspective

Transcription

1 Humboldt State University Digital Humboldt State University Congressional Apportionment Open Educational Resources and Data Congressional Apportionment: A Liberal Arts Perspective Charles M. Biles Humboldt State University, nia@suddenlink.net Follow this and additional works at: Recommended Citation Biles, Charles M., "Congressional Apportionment: A Liberal Arts Perspective" (2016). Congressional Apportionment This Articles and Essays is brought to you for free and open access by the Open Educational Resources and Data at Digital Humboldt State University. It has been accepted for inclusion in Congressional Apportionment by an authorized administrator of Digital Humboldt State University. For more information, please contact cyril.oberlander@humboldt.edu.

2 Congressional Apportionment: A Liberal Arts Perspective Charles M. Biles, Ph.D. Professor of Mathematics, Emeritus Humboldt State University Arcata, CA 20 September 2016

3 Preface Congressional apportionment of the United States House of Representatives is a popular topic in several liberal arts or general education mathematics textbooks. This work is produced in the belief that any illustrative applications of mathematics should be presented at the same high level as the mathematics. Since congressional apportionment is such a mainstay as a principle example of a mathematical, fixed resource problem, we believe that a more thorough presentation of the topic as it unfolded in American history is in order. Accordingly we dedicate five individual topics under the general heading of apportionment: averages and rounding, the basic divisor method, the quota method, the modified divisor method, and priority computational techniques. This work is intended as an open source endeavor. Comments, suggestions, critique, etc., are most welcome. The author is especially grateful to Professors Dale Oliver, Adam Falk, and Tim Lauck at Humboldt State University, and to Professors Dan Munton and John Martin at Santa Rosa Junior College for their collaborations and guest lecture invitations. I am also grateful for reviews from my dear wife Carolyn, and to friends Mickie York, Grace Kimura, and George Robinson. Special thanks goes to Professor David Lippman, Pierce College, Fort Steilacoom, Washington, for his encouragement and invitation to expand on his work on apportionment in the open-source text, Math and Society: A survey of mathematics for the liberal arts major, The sections in this work on quota method and the modified divisor methods are built on the foundation of the Apportionment chapter Math and Society. For additional open-resource materials, please visit Table of Contents Section 1: An Average Lecture 1 Section 2: Basic Divisor Methods 9 Section 3: Quota Methods 18 Section 4: Modified Divisor Methods 27 Section 5: Priority Techniques 32 Cover page image courtesy of Wikipedia Commons: i

4 Section 1: An Average Lesson An important aspect of education is to learn how to do research. A research project investigates a general field of interest along with its components. Research also includes the process of asking questions of interest and obtaining answers to some of those questions. In today s world the general public often just focuses the concept of research on getting answers. However, much preparation must be done before one can get useful answers. Research encompasses the entire process. 1. Background In conducting research one often comes across classic research problems. For example, how can one say something informative about a group when the individuals in the group are all different? This problem is often called the Problem of Variation or the Problem of Diversity. The word problem here is to be taken in a positive sense meaning a situation or reality. One should not take the word problem here in a negative sense meaning something disgusting or something to be avoided. Along with classic research problems come classic solutions; for example, denial. One way to face the problem of variation is to tell yourself, what variation? Ask yourself, What would all the members of the group look like if they all looked the same? This is precisely the kind of thinking behind the statistical concept of average and embodied in the English word typical. By appealing to a typical member of a group you can begin to talk about the members of a group in an organic, living sense. Of course denial is unrealistic if you stop there. But it can be an informative and useful beginning. To solve a problem (problem in the negative sense), ask yourself what the world would be like without the problem. For example, imagine a world without war, disease, or poverty. Imagine now a research project where you have a group of interest and also a question of interest that can be posed to each member of the group. Statisticians call such questions random variables because the members of the group give different answers to the question, hence the answers vary from member to member. The answers are random in that you do not know in advance the answer when the next member of the group presents itself. For example, my group of interest could be all the students enrolled in the current semester at Humboldt State University. Questions of interest (random variables) could include: What is your name? What level student are you (freshman, sophomore, junior, senior, graduate)? How old are you? What is your serum tetrahydrocannabinol level? Some questions of interest must be answered with a number obtained by counting or measuring something. Statisticians call such questions quantitative random variables. Statisticians calculate a mean in order to obtain an average for a list of numbers obtained from such variables. A mean of a set of numbers is an average for those numbers calculated in a way that preserves some rationale for those numbers. In this way, rather than being boggled by a bewildering array of numbers, one bypasses the variation and simply refers to this typical value. For example, HSU s website says that the average cost for a resident student at HSU is $23, Although costs differ from student to student, this average is a helpful reference when planning finances. 1 Accessed September

5 2. Means We now formalize the computational aspects of a mean. Recall that a mean of a set of numbers is an average calculated in a way that preserves some rationale for those numbers. Since numbers may be used for a variety of different purposes, accordingly there are a variety of different means. The three most common means are the arithmetic mean, the geometric mean, and the harmonic mean. We begin with the simplest kind of group, a group with only two members. So our current quest is: given two positive numbers a and b, how and why do we calculate the various means? Arithmetic Mean The arithmetic mean is the ordinary average of two numbers that is usually taught in the fifth grade: add the two numbers together and divide by 2. For example, the arithmetic mean of 4 and 16 is calculated by (4 + 16)/2 = 10. We denote the arithmetic mean (AM) of 4 and 16 by AM(4,16) = 10. In general, suppose a and b are two positive numbers. Then, the arithmetic mean of a and b is given by AM(a,b) = (aa+bb) 2 The arithmetic mean is an additive mean. Think of a and b as the length of two boards that we join linearly to make a longer board. The combined length is a + b. The thing we want to preserve is the combined length of the boards. Now consider making a final board with two components each having the same length. How long should each component be so that the result, a final board of length a + b, is preserved? For this rationale, each component needs to have length (a + b)/2; i.e., the arithmetic mean. Voila! Same result, but this time without the variation in the components. Hence, the general rationale for the arithmetic mean is as follows. Suppose I have two different positive numbers and add them together to get their sum (notice the variation in the numbers you are adding together). Total length a + b. a b Now, remove the variation. What would you use if both numbers were replaced by the same quantity but still gave the same sum? Answer: replace a and b by their arithmetic mean. aa + bb 22 aa + bb 22 Voila! Same result without the variation in the components. Total length a + b. Geometric Mean The geometric mean (GM) views the numbers a and b multiplicatively instead of additively. The geometric mean of two numbers is obtained by multiplying them together and then taking the square root of the resulting product. For example, GM(4,16) = 4 16 = 64 = 8. In general, the geometric mean of two positive numbers a and b is given by GM(a,b) = aa bb 2

6 The obvious question is, who would want to do such a thing? The thinking behind the geometric mean is similar to the thinking behind the arithmetic mean, but the geometric mean focuses on multiplication rather than addition. When I multiply 4x16 = 64, I notice that I am multiplying two different numbers to get the answer of 64. What if I wanted to replace the two different numbers by a single number to get the same answer? Hence,?x? = 64. To this end we need the square root of 64, which is 8. Suppose I made a rectangular laboratory space that is 4 ft. by 16 ft. The resulting working space is obtained by multiplication yielding a working area of 64 square feet. So, what would be the dimensions of the room which had the same work space, 64 square feet, but sides of equal length? For this rationale, we need the geometric mean. The arithmetic mean will not work. Hence, the general rationale for the geometric mean is as follows. Suppose I have two different positive numbers and multiply them to get their product (notice the variation in the numbers you are multiplying together). a Total area a b. b Now, remove the variation. What would you use if both numbers were replaced by the same quantity but still gave the same product? Answer: replace a and b by their geometric mean. aa bb Total area a b. aa bb Voila! Same result without the variation in the components. One can obtain a variety of applications of the geometric mean by googling applications of the geometric mean, especially applications of the geometric mean in business. Harmonic Mean An understanding of the harmonic mean begins with a classic formula from elementary school: distance = rate x time (in symbols, d = rt). For example, if I travel 240 miles in 6 hours, then my average speed is 40 miles per hour. Of course this does not mean that the car was in cruise control and you traveled 40 miles per hour the entire trip. What is so amazing about this averaging formula is that you don t even know what speeds are being averaged! Now suppose you make the return trip along the same route but it takes you 4 hours. Your average speed for the return trip is (240 miles)/(4 hours) = 60 miles per hour. So the arrival trip was made at 40 mph and the return trip at 60 mph. What was your average speed for the entire trip? How would you average 40 and 60 to obtain your average trip speed? To answer these questions, we need to determine 3

7 what rationale of the numbers we want to preserve. For an average total trip speed, the total distance is 480 miles (arrival plus return distances). The total time for the trip is 10 hours. Hence, the average speed for a 480 mile trip in 10 hours is 48 miles per hour. But, AM(40,60) = 50, so the arithmetic mean will not do. Also, GM(40,60) = = 2400 = , so the geometric mean won t do either. We need the harmonic mean. The harmonic mean (HM) of two positive numbers a and b is given by HM(a,b) = 2 1 aa + 1 bb = 2 aa bb aa+bb = aa bb (aa+bb)/2 = (GM)2 AM (*) Hence, the harmonic mean is a hybrid mean being equivalent to the square of the geometric mean divided by the arithmetic mean. In our example, HM(40,60) = 48. A formal explanation for the harmonic mean formula in (*) may be given as follows. Suppose we want to make a round trip from A to B with distance d between the two points. The total round-trip distance is then 2d. A d B Suppose that the time for the arrival trip from A to B is t 1. Since, in general, d = rt, then also r = d/t and t = d/r. So, the arrival trip speed, r 1, is calculated by r 1 = d/t 1. Thus, t 1 = d/r 1. Similarly, suppose that the time for the return trip is t 2. Then the return speed, r 2, is calculated by r 2 = d/t 2. Thus, t 2 = d/r 2. Now let r be the average speed and t be the total time for the whole round-trip. The round-trip distance is 2d and the round-trip time is given by t = t 1 + t 2. Then, for the entire round-trip, we have: 2d =rt 2d = r(t 1 + t 2 ) substitute: t = t 1 + t 2 2d = r dd rr 1 + dd rr 2 substitute: t 1 = d/r 1 and t 2 = d/r 2 2 = r 1 rr rr 2 cancel through by d r = 2 1 rr1 + 1 rr2 algebra r = HM(r 1,r 2 ) definition of harmonic mean We conclude that the average speed for the round-trip is the harmonic mean of the arrival trip speed and the return trip speed. 4

8 3. Averaging Suppose we are given two positive numbers, a and b. We want to report an average. What are our options? Here are five options that have applications. Only context and rationale can determine which option to use. min(a,b) max(a,b) AM(a,b) GM(a,b) HM(a,b) This is the minimum of a and b. It is the at least option. This is the maximum of a and b. It is the at most option. The arithmetic mean. The geometric mean. The harmonic mean. The following interesting mean inequality relationship always occurs for positive numbers where a < b. a = min(a,b) < HM(a,b) < GM(a,b) < AM(a,b) < max(a,b) = b For example, consider 4 and 16. Then, 4 = min(4,16) < HM(4,16) = 6.4 < GM(4,16) = 8 < AM(4,16) = 10 < max(4,16) = 16 Further, the only way that two of the quantities can be equal happens if and only if the two numbers a and b were the same to begin with. 4. Extending the Means Thus far we have provided mechanisms for calculating the AM, GM, and HM of two positive numbers. The formulas may be extended to include the special case that a = 0 and b > 0. The formula for the arithmetic mean applies immediately: AM(0,b) = 0+bb 2 = bb 2 Similarly for the geometric mean: GM(0,b) = 0 bb = 0 = 0. However, there is a situation with the harmonic mean. The formula for HM(a,b) involves 1/a. When a = 0 we have a problem. Fortunately, the algebraic simplification displayed for the harmonic mean formula (see formula * on the previous page) computes for a = 0 and b > 0: HM(0,b) = 2 0 bb 0+bb = 0 bb = 0 Although the formulas for the arithmetic, geometric, and harmonic means can be expanded to accommodate a = 0, the strict mean inequality does not carry over for a = 0. For a = 0 we have 0 = min(0,b) = HM(0,b) = GM(0,b) < AM(0,b) = b/2 < max(0,b) = b However we can expand the scope of the inequality to cover 0 a < b. In this event, we always have a = min(a,b) HM(a,b) GM(a,b) < AM(a,b) < max(a,b) = b 5

9 Finally we merely note that one can also extend calculating these means for more than two numbers. Suppose that a 1, a 2,, a n are n positive numbers. Then, AM(a 1, a 2,, a n ) = aa 1 + aa aa nn nn nn GM(a 1, a 2,, a n ) = aa 1 aa 2 aa nn HM(a 1, a 2,, a n ) = nn 1 aa1 + 1 aa aann However, in this course we will not have occasion to apply these means beyond two numbers. We also caution that the alternative forms for HM(a,b) given in (*) do not apply beyond two numbers. 5. An Application: Rounding Decimal Numbers Suppose that we have a positive decimal number that is not a whole number; i.e., a positive number with a genuine decimal part. Now, decimal fractions can be annoying. Hence, applying the classic research solution denial (what decimal fraction?) may provide convenience. Suppose that the integer part of our decimal number is n. We can then round the decimal down to n or round it up to n+1 to eliminate the decimal fraction. We will denote the rounding of a positive decimal number q by rnd(q). Note that rnd(q) will be either n or n+1. There are five ways, depending on the application, to round a positive decimal number, q, with integer part n. In particular, rnd(q) = 1. min(n, n+1) = n This is the round down option. 2. max(n, n+1) = n + 1 This is the round up option. 3. n + 1 q > n + ½ q > AM(n, n+1) The usual method of rounding a decimal. 4. n + 1 q > GM(n, n+1) The geometric mean option. 5. n + 1 q > HM(n, n+1) The harmonic mean option. To illustrate, let s work through an example. We apply the five methods to illustrate how to round Should I round up to 3 or round down to 2? 1. rnd(2.437) = 2 min option: round down 2. rnd(2.437) = 3 max option: round up 3. rnd(2.437) = 2 usual rounding method: < AM(2,3) = rnd(2.437) = 2 geometric mean option: < GM(2,3) = rnd(2.437) = 3 harmonic mean option: > HM(2,3) = 2.4 6

10 Exercises 1. Find the following averages. a. min(7,30) b. max(7,30) c. AM(7,30) d. GM(7,30) e. HM(7,30) 2. Compute the arithmetic, geometric, and harmonic mean for each pair of numbers. a. 3 and 48 b. 4 and 7 c. 8 and 20 d. 8 and 9 e. 6 and Complete the table by rounding each of the given decimal numbers using the indicated method. Decimal Method min max AM GM HM 4. Sabermetrics is the mathematical and statistical analysis of baseball records. 2 Most baseball fans are familiar with routine measures such as batting average, on base percentage, and even the power batting average where each hit is weighted by the number of bases reached by the hit. Some baseball aficionados find the power-speed number for a player a useful statistic to measure a player s clutch offensive productivity. The power-speed number is the harmonic mean of a player s home runs and stolen bases. The lifetime major league leaders for the power-speed index are the Giants legend Barry Bonds (613.90) and the A s Ricky Henderson. 3 a. Lifetime, Ricky Henderson hit 297 home runs and had 1406 stolen bases. What was Ricky Henderson s lifetime power-speed number? b. Ricky Henderson played 14 years for the Oakland A s. During this time with the A s, Henderson hit 167 home runs and stole 867 bases. What was Henderson s power-speed number at Oakland? 5. Compute the arithmetic, geometric and harmonic mean for each set of numbers. a. 1, 3, 5. b. 10, 12, 15, 18,

11 6. Use the adjacent figure to show that h is the geometric mean of a and b. The three indicated angles in the figure are all right angles. [Hint: The Pythagorean Theorem can come in handy.] h a b 7. (Challenge) Consider the circle shown at right. Let AAAA = a and BBBB = b. The center of the circle is at C. FFFF AAAA and BBBB. CCCC F a. Verify that AM(a,b) = AAAA. b. Verify that GM(a,b) = BBBB. c. Verify that HM(a,b) = EEEE. d. Explain why the diagram is a proof without words for the mean inequality. D C E B A 8

12 Section 2: Basic Divisor Methods The Congressional Apportionment Problem Based on the Census : no political problem is less susceptible of a precise solution than that which relates to the number most convenient for a representative legislature, James Madison 4 The Federalist 55 The Congressional Apportionment Problem is an engaging application of mathematics to an ongoing problem in American history. The cast includes many well-known characters including George Washington, Thomas Jefferson, Alexander Hamilton, John Quincy Adams, Daniel Webster, James K. Polk, and Franklin Roosevelt. The problem is deceptively easy to state. Congressional Apportionment Problem (CAP). Determine the number of seats each state gets in the United States House of Representatives based on the decennial census. Any doubts about the meaning of the problem are quickly resolved by considering the current situation as displayed in Figure

13 Figure 1. The congressional apportionment map based on the 2010 census. 5 There are currently 50 states in the Union and 435 seats in the U. S. House of Representatives. The apportionment population for the nation was 309,183,463 based on the 2010 census. Thus, each congressperson represents about 711,000 people. The distribution of seats is made on the basis of population. California, the most populous state, has 53 seats, followed by Texas with 36. Seven states have the minimum representation of 1 seat each: Alaska, Montana, Wyoming, North Dakota, South Dakota, Vermont, and Delaware. 1. Constitutional Guidelines The U. S. Constitution specifies the basis for representation immediately following the Preamble. Article I, Section 1, states that all law making powers are vested in Congress consisting of a Senate and a House. Representation in the Senate is based on geography: each state has two senators. Guidelines for the House are minimal but substantial. Representation is based on population as determined by a decennial census. The Constitution also sets criteria for the minimum and maximum House size. Each state must have at least one representative. Further, the House size shall not exceed one for every thirty Thousand. This means that in general a congressperson may represent or more people, but not less. The initial congressional apportionment is specified in Article I, Section 2, and is known as the constitutional apportionment. It was based on the framers estimates of the state populations in This constitutional apportionment would remain in effect until reapportionment based on the first census. The first Congress admitted Vermont as the fourteenth State in the Union. It also authorized the first U. S. census which began in Accordingly, each year ending in a zero is a census year. Congress also passed an enabling act anticipating statehood for Kentucky in the near future. Hence, the first census involved fifteen states. The census report was submitted The U.S. Constitution: Article I Section 1. All legislative Powers herein granted shall be vested in a Congress of the United States, which shall consist of a Senate and House of Representatives. Section 2. The House of Representatives shall be composed of Members chosen every second Year by the People of the several States, and the Electors in each State shall have the Qualifications requisite for Electors of the most numerous Branch of the State Legislature. No Person shall be a Representative who shall not have attained to the Age of twenty five Years, and been seven Years a Citizen of the United States, and who shall not, when elected, be an Inhabitant of the State in which he shall be chosen. Representatives and direct Taxes shall be apportioned among the several States which may be included within this Union, according to their respective Numbers, which shall be determined by adding to the whole Number of free Persons, including those bound to Service for a Term of Years, and excluding Indians not taxed, three fifths of all other Persons. The actual Enumeration shall be made within three Years after the first Meeting of the Congress of the United States, and within every subsequent Term of ten Years, in such Manner as they shall be Law direct. The Number of Representatives shall not exceed one for every thirty Thousand, but each State shall have at Least one Representative; and until such enumeration shall be made, the State of new Hampshire shall be entitled to chuse three, Massachusetts eight, Rhode-Island and Providence Plantation one, Connecticut five, New-York six, New Jersey four, Pennsylvania eight, Delaware one, Maryland six, Virginia ten, North Carolina five, South Carolina five, and Georgia three. 10

14 to Congress by President George Washington on 28 October To understand where we are today, we go back to this first census. 2. Re-apportionment based on the 1790 Census Congress received the census report on a Friday and went to work on apportionment the following Monday. The main view was that the Senate represented the States and that the House represented the People. Congressmen wanted maximum representation for the people. They began with the question, how many people should a congressman represent? Their answer: The answer is known as the ratio of representation (or more simply the ratio), the constituency, or the divisor. Accordingly they divided into the population of each state to determine how many representatives each state deserved. They took only the integer part of the answer. They felt that any fractional remainder was not enough to justify an additional representative. The results are displayed in Figure 2, House Bill main column. 6 It took the House just one month to finalize their bill which was then sent to the Senate for concurrence. The Senate felt that was a better answer to the question, how many people should a congressman represent? They applied the same methodology as the House but used as the divisor. The results are displayed in Figure 2, Senate Bill main column. It took the Senate one month to finalize its bill. However, neither chamber of Congress would accede to the other resulting in an impasse. To break the gridlock Congress needed to come up with some out-of-the-box thinking. A new approach was offered by Federalists in the House. They suggested starting with the House size rather than the constituency. Once the House size is known, then the thinking is obvious: if a state has 10% of the population, then it should have 10% of the seats in the House. Accordingly they presented the Rule of Three to calculate each state s quota: ssssssssss pppppppppppppppppppp qqqqqqqqqq = (HHHHHHHHHH ssssssss) nnnnnnnnnnnnnnnn pppppppppppppppppppp Census House Bill Senate Bill State Population d = Seats d = Seats CT DE GA KY MD MA NH NJ NY NC PA RI SC VT VA US Figure 2. The 1790 Census data with the first House and Senate apportionment bills; d represents the constituency divisor. The term Rule of Three highlights that to compute the quota one must use three things: the House size, the state population, and the national population. 6 The census figures are taken from Michel Balinski and H. Peyton Young, Fair Representation: Meeting the Ideal of One Man, One Vote, 2nd, Washington, D.C.: Brookings Institution Press, 2001: 158. These are the final and corrected census figures, not the original data submitted by President Washington. Initially Congress had to deal with incomplete returns from South Carolina and some corrections made on the fly. However, the results presented here are consistent with the results historically obtained. For a detailed account of the history, see Charles Biles, Congressional Apportionment Based on the Census 1790, available as an open resource download from 11

15 Federalists in the House used this idea to test the result of the House and Senate bills. Note in particular that the House size was never used in making the House and Senate bills the House size was merely a result of the apportionment methodology. But, once a bill is finalized, then the Rule of Three can be applied to the resulting House size. Census Figure 3 displays the result of testing the House and Senate bills with the Rule of Three. The House bill created a House with 112 seats. The House Bill, Quota column, displays each state s fair share based on a House size of 112. The concept of quota contains an intrinsic rule of fairness known as the quota rule. For example, Connecticut s fair share of 112 seats is However, fractional seats are impossible; hence, Connecticut s fair share is at least 7 but no more than 8. In general, the quota rule asserts that a state s fair share must be the quota rounded down or rounded up. The Rule of Three exposes a quota rule violation for Virginia. Virginia s fair share quota of 112 seats is , yet the House bill gives Virginia 21 seats. This problem became an eventual deal-breaker for the House bill. State Population d =30000 Seats Quota d =33000 Seats Quota CT DE GA KY MD MA NH NJ NY NC PA RI SC VT VA US Interestingly the Senate bill has no quota rule violation. Quota rule violations are possible for any apportionment using a constituency approach. A quota rule violation does not have to occur using a given divisor, as the Senate bill verifies, but it may occur as the House bill verifies. Although free of any quota rule violation, the Senate bill has an annoying feature from the viewpoint of the quota. Virginia s fair share of 105 seats is , yet the Senate bill gives Virginia 19 seats. In contrast, Delaware s fair share of 105 seats is 1.613, yet the bill gives Delaware only 1 seat. Is it really fair that a state with a lower decimal quota is rounded up over a state with a higher decimal quota? This apparent favoritism is the result of the round down criterion that was applied to the quotient = (state population)/divisor. Both the House and Senate bills rounded all quotients by rounding down. Using the rounding down procedure on all decimal quotients may lead to a biased favoritism in the quota. Such resulting favoritism always favors a larger state over a smaller state. With the discovery of these two flaws in the constituency approach to apportionment, Federalists felt that they had leverage to advocate their plan based on a House size approach. Federalists advanced the idea that initiating apportionment on the constituency question got things off on the wrong foot as evidenced by the results. Instead of asking, how many people should a congressman represent, we start with asking, what should be the size of the House? To advance their plan, Federalists advocated maximum representation for the people. They also used the figure from the Constitution but began by dividing into the national population which yielded Accordingly, following the constitutional constraint that the size of the House may not exceed one in thirty-thousand, the maximum allowable House size is 120. Federalists then asked, what is each state s fair share of 120? The results are shown in Figure House Bill Senate Bill Figure 3. Quota Rule analysis of the first House and Senate apportionment bills.

16 The subsequent method is known as Hamilton s method in today s literature. The method first applies the Rule of Three to obtain each state s fair share quota for the given House size of 120 members. The method then allocates the lower quota (the quota rounded down) to each state. This distributes 111 of the 120 seats. There then remains 9 seats to distribute among the 15 states. These 9 seats are awarded to the 9 states with the largest decimal component in the quota. The decimal components of the quota may be thought of as a priority list. Accordingly, New Jersey with decimal component.96 is awarded the 112 th seat and New Hampshire with decimal component.71 is awarded the 120 th seat, completing the distribution of the 120 seats. The result had several remarkable advantages going for it. First, there are no quota rule violations; in fact, there can t be any quota rule violations since Census Hamilton's Method State Population h = 120 Quota Lower Q Appt CT DE GA KY MD MA NH NJ NY NC PA RI SC VT VA US Figure 4. The first apportionment bill passed by Congress. the distribution of seats is founded on the quota. Automatically, each state is given either the quota rounded down or rounded up as need be. Second, there cannot be any biased favoritism since additional seats are distributed according to largest fractions. Hence, the objections to the House and Senate bills were automatically overcome. Further, by a freak happenstance of the data, the seats allocated (Figure 4, Hamilton s Method, Appt column) correspond to an ordinary rounding of the quota. Even better, each state that was given an additional seat beyond the lower quota had a decimal fraction greater than.7 and each state given the lower quota had a decimal fraction less than.4. With all this going for it, this bill broke the House-Senate gridlock and became the first apportionment bill passed by Congress. On 26 March 1792, five months after receiving the census, Congress sent the bill to President Washington for his approval and signature. President Washington vetoed the bill. The veto is significant for three reasons. It was the first presidential veto in U.S. history. It was the only veto of Washington s first administration. Washington justified his veto based on his interpretation of the Constitution. The House size of 120 yields /120 = when applied to the U.S. population as a whole. But, when applied to Connecticut, /8 = Washington insisted that the constitutional constraint that the size of the House shall not exceed one for every thirty Thousand must be satisfied by each state individually, not just the nation as a whole. After Washington s veto, Congress quickly passed the original Senate bill which Washington signed on 14 April Basic Divisor Methods The debate over re-apportionment based on the 1790 census displayed two approaches to the congressional apportionment problem: a constituency approach and a House size approach. A constituency approach is based on the question, how many people should a congressman represent? A House size approach is based on the question, how many seats should there be in the House? 13

17 The method used to construct the original Senate bill that eventually became the first apportionment act set precedent and was used for the next five censuses (see Figure 5). The method is called a basic divisor method and is based on a constituency approach. It involves a 3-step algorithm: Step 1. Step 2. Step 3. Determine how many people a congressman should represent. Answer: d. Calculate each state s quotient: quotient = (state population)/d. Round the quotient to obtain the state s apportionment. 1790: s = 15, d = h = : s = 16, d = h = : s = 17, d = h = : s = 24, d = h = : s = 24, d = h = : s = 26, d = h = 223 Figure 5. A basic divisor method applied to the first six censuses; s represents the number of states, d the divisor, and h the House size. The apportionment act based on the 1790 census used a basic divisor method in which each state s quotient was rounded down. We refer to this method as Jefferson s method, or more completely, Jefferson s basic divisor method. Jefferson s method was used for apportionments based on the census from 1790 to 1830, inclusive. Flaws with Jefferson s method were evident from the start, but new quota rule violations demanded attention. Alternate proposals for rounding the decimal quotient surfaced. During the 1830 censusbased apportionment debates, Daniel Webster, chair of the Senate apportionment committee, received letters from John Quincy Adams, a representative from Massachusetts, and James Dean, a mathematics professor at the University of Vermont. Thinking about alternatives proposed by Adams and Dean, Webster devised his own. Thus, four variations of the basic divisor method, all dealing with how to round a decimal (the quotient), were available to Webster. Jefferson: round down. Adams: round up. Dean: round down or up depending on which option gives a state s constituency closer to the divisor. Webster: round normally. For apportionment based on the 1830 census Congress used the precedent Jefferson method. However, other methods were now on the table. Apportionment based on the 1840 census used a constituency approach with the divisor and, for the first time, Webster s method for rounding the quotient. This resulted in a House with 233 members. It was the only time in U.S. history that the House size decreased as a result of the decennial census-based reapportionment process. 4 Basic Divisor Methods Step 3 in the basic divisor method involves rounding the quotient. During the debates based on the 1830 census, Daniel Webster had four proposals for how to round the quotient. Jefferson s, Adams s, and Webster s rounding criteria are easily familiar. We now take a closer look at Dean s method by looking at an example. In 1830 the US population was 11,931,578. Consider: constituency = 50,000 people. Vermont s population: 280,657. Vermont s quotient: 280,657/50,000 =

18 At this point, Jefferson apportions 5 seats to Vermont; Adams, 6 seats. With 5 seats the constituency is 280,657/5 = 56,131. With 6 seats the constituency is 280,657/6 = 46,776. A constituency of 46,776 is closer to the target constituency of 50,000; hence, Dean awards Vermont 6 seats. Dean s and Webster s methods are similar in their thinking. Webster s method involves rounding the decimal quotient normally; i.e., if the decimal fraction is less than.5, then round down, otherwise round up. Denote the quotient by q and let n be the integer part of q. Then, q rounded down is n and q rounded up is n+1. Rounding q normally is equivalent to the criterion: round up if and only if q is greater than the arithmetic mean of the round down, round up options; i.e., q > AM(n,n+1). Dean s method is mathematically equivalent to the criterion: round up if and only if q is greater than the harmonic mean of the round down, round up options; i.e., q > HM(n,n+1). 7 The four rounding options may be expressed in terms of the quotient as follows. A state s apportionment is obtained by rounding the quotient, q, where you round up if and only if Jefferson: q > max(n,n+1) Since this can t happen, always round down. Adams: q > min(n,n+1) Since this always happens, always round up. Webster: q > AM(n,n+1) Round normally. Dean: q > HM(n,n+1) Round by closest constituency. For illustration, we now apply these four basic divisor methods to the 1810 census (see Figure 6). Although there is a lot of data in the displayed spreadsheet, one can quickly grasp the main elements. First, Census 1810 lists the 17 states with their populations. Second, Congress used a constituency approach with a congressman representing people. Third, is divided into each state s population to determine each state s Quotient. Fourth, the quotient is rounded applying the four variations: Jefferson, Adams, Webster, and Dean. With the fixed constituency of 35000, the four rounding methods each lead to a different House size. Since Jefferson rounds all quotients down, this method produces the smallest House with 181 members. Since Adams rounds all quotients up, this method produces the Census 1810 d = State Population Quotient Jefferson Webster Dean Adams CT DE GA KY MD MA NH NJ NY NC OH PA RI SC TN VT VA US Figure 6. The 1910 census. largest House size with 198 members. Since Webster and Dean round some states up and other states down, they produce a House with an intermediate size. Note that Dean s method produces a House with one more member than Webster s method. The involved state is Connecticut whose quotient is 7 For the mathematical derivation of these equivalencies, see Charles Biles, Congressional Apportionment Based on the Census : 50-53; available as an open-source download from 15

19 Rounding normally, Webster rounds the quotient down and awards Connecticut 7 seats. However, Dean s rounding criterion is to round up if the quotient is larger than the harmonic mean of the round-down, round-up options. Here, HM(7,8) = Thus, Dean awards Connecticut 8 seats. Hence, apportionment methodology matters. The criterion for rounding the quotient matters. Although different methods can produce the same result, they may all produce different results as exhibited by congressional apportionment based on the 1810 census. Re-apportionment resulting from the first six censuses was accomplished using a basic divisor method. The first five re-apportionments used Jefferson s method. The sixth, based on the 1840 census, used Webster s method. However, the basic divisor method became subject to serious political manipulations and Congress looked for an alternative to eliminate gaming the system. Accordingly, in the apportionment debate based on the 1850 census Congress abandoned the basic divisor approach and applied a House size approach. This transformed the congressional apportionment problem into a mathematical apportionment problem. Exercises 1. Consider the 1790 census. A. Create a spreadsheet showing the effects of applying the four basic divisor methods (Jefferson, Adams, Webster, Dean) to the 1790 census using the divisor B. Do the four methods lead to different results? 2. Repeat Exercise 1 for the 1800 census using the divisor Repeat Exercise 1 for the 1820 census using the divisor Repeat Exercise 1 for the 1830 census using the divisor Repeat Exercise 1 for the 1840 census using the divisor The state of Delaware has three counties: Kent, New Castle, and Sussex. 8 A. Complete the following table to apportion the Delaware House of Representatives using the indicated basic divisor method (see Figure 6 for illustration). Census 2010 d = County Population Quotient Jefferson Webster Dean Adams Kent New Castle Sussex Delaware Exercises 6 and 7 are adapted from Mathematics and Society: A Survey of mathematics for the liberal arts major, 2012: Available from 16

20 B. Repeat A. using a divisor of C. Repeat A. using a divisor or The state of Rhode Island has five counties: Bristol, Kent, Newport, Providence, and Washington. Complete the following table to apportion the Delaware House of Representatives using the indicated basic divisor method (see Figure 6 for illustration). Census 2010 d = County Population Quotient Jefferson Webster Dean Adams Bristol Kent Newport Providence Washington Rhode Island

21 Section 3: Quota Methods Congressional apportionment based on the 1850 census saw a major evolution in methodology. Apportionment acts based on the census were founded on the basic divisor method. In addition to problems caused by occasional quota rule violations, politicians became adept at gaming the system. In a single day during debates based on the 1840 census, 59 different proposals for a divisor were made in the House, including numbers such as 62279, 59241, and The House bill settled on On one day in the Senate there were 27 proposals for a divisor. The Senate settled on The apportionment act of 1842 finally specified and Webster s method. 9 In 1850 Representative Samuel Vinton (Whig-Ohio) proposed a bill to stop the political gamesmanship. His plan established the House size in advance of the census and then set congressional apportionment on automatic pilot. The apportionment act of 1850 specified a House size of 233 with distribution calculated by the Census Office using Hamilton s method. The Quota Method Algorithm The apportionment act of 1850, also known as the Vinton act, transformed congressional apportionment into a modern day mathematical apportionment problem. In today s mathematics literature, apportionment is a fixed-resources problem. Formally, apportionment is the problem of distributing a fixed number of units among two or more groups. Arguably the major problem in the world today is the equitable distribution of resources. Apportionment is thus applicable to a wide range of problems. There are several guidelines we must observe to create equitable solutions to apportionment problems. Apportionment Guidelines 1. The units being distributed can exist only in whole numbers. 2. We must use all of the units being distributed and we cannot use any more. 3. Each group must get at least one of the units being distributed. 4. The number of units assigned to each group should be at least approximately proportional to the assets of the group. (Exact proportionality isn t possible because of the whole number requirement, but we should try to be close, and in any case, if Group A is larger than Group B, then Group B shouldn t get more units than Group A.) These guidelines transform into a 4-step quota method algorithm. Step 1. Step 2. Determine the number of resource units to be distributed. Determine each group s quota by applying the rule of three: Group Quota = (Resource Units) GGGGGGGGGG AAAAAAAAAAAA TTTTTTTTTT AAAAAAAAAAAAAA Step 3. Allocate each group its lower quota (the quota rounded down). 9 Charles Biles, Congressional Apportionment Based on the Census : Available on-line at 18

22 Step 4. Distribute any remaining resource units according to a priority list based on each group s assets. For congressional apportionment of the U.S. House of Representatives, the quota method algorithm uses a state s population as the measure of its assets since the Constitution mandates that apportionment shall be based on population. Step 1. Select the House size, h. Currently, h = 435. Step 2. Calculate each state s quota: Quota = h ssssssssss pppppppppppppppppppp nnnnnnnnnnnnnnnn pppppppppppppppppppp Step 3. Allocate each state its lower quota. Step 4. Distribute any remaining seats according to a priority list based on each state s population. We examine three ways of completing Step 4: Hamilton, Lowndes, and Hill. Hamilton s method is what is usually applied and initially appeared during the apportionment debates based on the 1790 census. Lowndes s method was proposed during the debates based on the 1820 census. 10 Hill s method was proposed during the debates based on the 1910 census. 11 Almost always Step 3 distributes most but not all of the House seats. Further, the number of remaining seats for distribution is less than the number of states. Hence, to complete Step 4 we need to create a priority list to determine which states get the remaining seats. The priority lists are calculated as follows. Hamilton: the decimal fraction of the quota. Lowndes: lower quota constituency = (state population)/(lower quota) Hill: (state population)/ LL UU where L is the lower quota and U is the upper quota. Hamilton s Method We first saw Hamilton s method in the congressional apportionment debates based on the census of The original congressional bill, which President George Washington vetoed, was based on House size 120 with apportionment based on Hamilton s method where additional seats were allocated by largest fractions. We now work through two examples to highlight the details. Example 1 Delaware has three counties: Kent, New Castle, and Sussex. The Delaware House of Representatives has 41 members. We will apportion this representation along county lines (which is not required) by applying Hamilton s method. The populations of the counties are based on the 2010 census. Step 1. Delaware s House size is Charles Biles, Congressional Apportionment Based on the Census : Available on-line at 11 Charles Biles, Congressional Apportionment Based on the Census : Available on-line at 19

23 Step 2. Determine each county s quota by applying the rule of three. Census 2010 Hamilton's Method County Population Quota Lower Seats Kent New Castle Sussex Delaware Step 3. Step 4. Determine each county s lower quota (quota rounded down). Census 2010 Hamilton's Method County Population Quota Lower Seats Kent New Castle Sussex Delaware Step 3 distributes 40 of the 41 seats. The remaining seat goes to the county with the largest decimal part, New Castle. Census 2010 Hamilton's Method County Population Quota Lower Seats Kent New Castle Sussex Delaware Example 2 We apply Hamilton s method to apportion the 75 seats of Rhode Island s House of Representatives among its five counties based on the census of Step 1: The House size is 75. Step 2: Determine each county s quota by applying the rule of three. Census 2010 Hamilton's Method County Population Quota Lower Seats Bristol Kent Newport Providence Washington Rhode Island

24 Step 3: Determine each county s lower quota. Census 2010 Hamilton's Method County Population Quota Lower Seats Bristol Kent Newport Providence Washington Rhode Island Step 4. We need 75 representatives and we only have 72, so we assign the remaining three, one each, to the three counties with the largest decimal parts, which in decreasing order of priority are Newport, Kent, and Providence. Census 2010 Hamilton's Method County Population Quota Lower Seats Bristol Kent Newport Providence Washington Rhode Island Note that even though Bristol County s decimal part is greater than.5, it isn t big enough to get an additional representative because three other counties have greater decimal parts. Hamilton s method obeys an intuitive rule of fairness called the quota rule. In general the quota is a quantity that represents each group s fair share of the resource units based on its assets. Hence, each group deserves at least its lower quota, but no more than its upper quota. In terms of congressional apportionment, the quota rule states the following. Quota Rule. The final number of representatives a state gets should be within one of that state s quota. Since we re dealing with whole numbers for our final answers, this means that each state should be allocated either its lower quota or upper quota. Lowndes s Method Lowndes s method differs from Hamilton s method in Step 4 of the quota method algorithm. Instead of using the quota s largest fractions to determine priority for an additional seat, Lowndes uses the constituency obtained from the lower quota. To illustrate, we rework Hamilton s Example 1 and Example 2 with Lowndes s method. Since Steps 1-3 are the same for both methods, we only present Step 4. 21

25 Example 1: Delaware Step 4: Calculate each county s constituency based on the lower quota; i.e., country constituency = (county population)/(lower quota). Census 2010 Lowndes's Method County Population Quota Lower Constituency Seats Kent New Castle Sussex Delaware In contrast to Hamilton, Lowndes gives the additional seat to Kent instead of New Castle. As a result, Kent s county constituency is reduced from to Example 2: Rhode Island Step 4: Calculate each county s constituency based on the lower quota; i.e., country constituency = (county population)/(lower quota). Census 2010 Lowndes's Method County Population Quota Lower Constituency Seats Bristol Kent Newport Providence Washington Rhode Island Giving each county its lower quota distributes 72 of 75 seats. The remaining three seats are given to the three counties with the highest constituency based on the lower quota. These counties in descending order are Bristol, Newport, and Kent. Note that Lowndes gives a seat to Bristol at the expense of Providence in comparison with Hamilton. Hill s Method Joseph Hill was a statistician in the Census Bureau at the time of the 1910 census. Hill was concerned about the paradoxes produced by Hamilton s method. He thought that he could rescue Hamilton s method by a different method of allocating an additional seat in Step 4 of the quota method algorithm. Hill recommended that the priorities for Step 4 be computed by (state population)/ LL UU where L is the lower quota and U is the upper quota. Recall that LL UU is the geometric mean of L and U. To illustrate, we rework Hamilton s Example 1 and Example 2 with Hill s method. Since Steps 1-3 are the same for both methods, we only present Step 4. 22

26 Example 1: Delaware Step 4: Calculate each county s Hill priority: population/ LL UU. Note that in this example Hill agrees with Hamilton. Example 2: Rhode Island Census 2010 Hill's Method County Population Quota Lower Priority Seats Kent New Castle Sussex Delaware Step 4: Calculate each county s Hill priority: population/ LL UU. Census 2010 Hill's Method County Population Quota Lower Priority Seats Bristol Kent Newport Providence Washington Rhode Island Giving each county its lower quota distributes 72 of 75 seats. The remaining three seats are given to the three counties with the highest Hill priority values. These counties, in descending order, are Bristol, Newport, and Kent. Note that Lowndes and Hill agree and, in contrast to Hamilton, give a seat to Bristol at the expense of Providence. Paradoxes A quota method initially seems quite fair since it is based on the quota seen as a group s fair share based on the group s assets. The eventual deal-breaker for the basic divisor method was the appearance of quota rule violations and the fact that it became subject to extensive political gamesmanship. The deal-breaker for quota methods turned out to be unexpected mathematical paradoxes. The three most prominent paradoxes are the Alabama paradox, the New States Paradox, and the Population paradox. The Alabama Paradox is named for an incident that happened during the apportionment that took place after the 1880 census. A similar incident happened ten years earlier involving the state of Rhode Island, but the paradox is named after Alabama. Based on the 1880 census William Walker, Director of the Census, noted a strange anomaly: at House size 299 Alabama had 8 seats, but at House size 300 Alabama got only 7 seats. The question arose, how can an increase in resource units result in a 23

27 decreased group share? This motivated Congress to pay more attention to the results of Hamilton s method. The deal breaker for Hamilton s method came during debates based on the 1910 census. The Alabama paradox shined a spotlight on Colorado at House size 357. Colorado received two seats at House size 357 but three seats for all other House sizes between 350 and 400. The Alabama paradox also spotlighted West Virginia with 5 seats at House size 351, only four seats at House size 352 and 353, and back up to 5 seats at House size 354. The paradox was so vivid for Maine that several congressmen became convinced that the Hamilton method was flawed beyond acceptance. Maine received three seats for House sizes , 386, and but four seats for House sizes , , and The New States Paradox was noticed when Oklahoma became the 46 th state in Oklahoma had enough population to qualify for five representatives in Congress. These five representatives were assigned to Oklahoma in addition to the 386 already apportioned to the 45 existing states. But statisticians noted a strange phenomenon. When Hamilton s method is applied to the 1900 census with Oklahoma included, the resulting apportionment is the same except that Maine gains a representative at the expense of New York. Why should the addition of a new state affect two others? The Population Paradox happened between the apportionments after the census of 1900 and of In those ten years, Virginia s population grew at an average annual rate of 1.07%, while Maine s grew at an average annual rate of 0.67%. Virginia started with more people, grew at a faster rate, grew by more people, and ended up with more people than Maine. By itself, that doesn t mean that Virginia should gain representatives or Maine shouldn t, because there are lots of other states involved. But Virginia ended up losing a representative to Maine. But things can even be worse. It is possible that a state may gain in population yet lose a seat to a state that drops in population. Exercises Apply the methods of Hamilton, Lowndes, and Hill, to apportion Exercises A college offers tutoring in Math, English, Chemistry, and Biology. The number of students enrolled in each subject is listed below. If the college can only afford to hire 15 tutors, determine how many tutors should be assigned to each subject. Math: 330 English: 265 Chemistry: 130 Biology: Reapportion the previous problem if the college can hire 20 tutors. 3. The number of salespeople assigned to work during a shift is apportioned based on the average number of customers during that shift. Apportion 20 salespeople given the information below. Shift Morning Midday Afternoon Evening Average number of customers 4. Reapportion the previous problem if the store has 25 salespeople. 24

28 5. Three people invest in a treasure dive, each investing the amount listed below. The dive results in 36 gold coins. Apportion those coins to the investors. Alice: $7,600 Ben: $5,900 Carlos: $1, Reapportion the previous problem if 37 gold coins are recovered. 7. A small country consists of five states, whose populations are listed below. If the legislature has 119 seats, apportion the seats. Allora: 810,000 Bella: 473,000 Cara: 292,000 Dulce: 594,000 Esotico: 211, A small country consists of six states, whose populations are listed below. If the legislature has 200 seats, apportion the seats. Acadie: 3,411 Beau: 2,421 Cognac: 11,586 Dordogne: 4,494 Été: 3,126 Fleur: 4, A small country consists of three colorful states with populations as follows: Azure, 6,000; Brown, 6,000; Crimson, 2,000. A. Apply Hamilton s method to apportion a 10-seat legislature. B. If the legislature grows to 11 seats, use Hamilton s method to apportion the seats. C. Which apportionment paradox does this illustrate? 10. A state with five counties has 50 seats in their legislature. Using Hamilton s method, apportion the seats based on the 2000 census, then again using the 2010 census. Which apportionment paradox does this illustrate? County 2000 Population 2010 Population Jefferson 60,000 60,000 Clay 31,200 31,200 Madison 69,200 72,400 Jackson 81,600 81,600 Franklin 118, , A school district has two high schools: Lowell, serving 1715 students, and Fairview, serving The district could only afford to hire 13 guidance counselors. A. Determine how many counselors should be assigned to each school using Hamilton's method. B. The following year, the district expands to include a third school, serving 2989 students. By increasing the number of counselors proportionally, how many additional counselors should be hired for the new school? C. After hiring that many new counselors, the district recalculates the reapportion using Hamilton's method. Determine the outcome. D. Does this situation illustrate any apportionment issues? 25

29 12. A small country consists of four states whose populations are listed below. The legislature has 116 seats. Apportion the seats using Hamilton s method. Does this illustrate any apportionment issues? Altura: 33,700 Baja: 559,500 Coast: 141,300 Desert: 89,100 26

30 Section 4: Modified Divisor Methods Apportionment acts based on the census were fabricated using a basic divisor method: Jefferson for and Webster for Apportionment acts based on the census were based on Hamilton s quota method. Congress abandoned the basic divisor method based on quota rule violations and excessive political gamesmanship. Congress wanted to abandon the quota method during the debates based on the 1900 census. However, there are only two approaches for the congressional apportionment problem: the constituency approach and the House size approach. The constituency approach is initiated by the question, how many people should a congressman represent? The House size approach is initiated by the question, how many seats should be in the House? The constituency approach naturally leads to a basic divisor method. The House size approach naturally leads to a quota method. What should Congress do now that both methods seemed unsatisfactory? Congress searched for an improved method, one that would preserve the ease and fairness of the two known approaches but would suppress the resulting objections that could occur. Consequently they produced a hybrid method that incorporated both approaches. Modified Divisor Methods A modified divisor method starts with the House size approach by setting the House size. The House size is then set aside as a goal, the predetermined answer for a basic divisor method. A basic divisor method needs a divisor, the ratio of representation. The divisor process starts by calculating the standard divisor defined as the total population divided by h. In congressional apportionment of the U. S. House of Representatives the standard divisor represents the constituency of an average congressional district based on the given House size and the national population. In general, the result is a 5-step modified divisor method algorithm. Step 1. Determine the House size, h. Step 2. Calculate the standard divisor: (total population)/h. Step 3. Calculate each state s quotient using the given divisor. Step 4. Round each state s quotient to obtain each state s apportionment. Step 5. If the apportionments all sum to h, then DONE; ELSE, modify the divisor and GO TO Step 3. Note that the algorithm begins by setting the House size. Then, the House size is set aside as the answer for a basic divisor method. The standard divisor supplies us with a ratio of representation for a basic divisor method. Normally, using the standard divisor, the resulting House size does not match h. If the resulting House size is smaller than h, then the divisor was too big; hence, decrease the divisor and return to Step 3. If the resulting House size is larger than h, then the divisor was too small; hence, increase the divisor and return to Step 3. A method of rounding needs to be adopted for Step 4. Thus far we have seen four rounding methods for congressional apportionment: Jefferson (round down), Adams (round up), Webster (round normally), and Dean (round by closest constituency). Accordingly we have Jefferson s, Adams s, Webster s, and Dean s modified divisor methods. Recall that Webster s method is mathematically equivalent to round the quotient up if and only if the quotient is greater than the arithmetic mean of the round down, round up options. Similarly, Dean s method is mathematically equivalent to round the 27

31 quotient up if and only if the quotient is greater than the harmonic mean of the round down, round up options. With the methods of Jefferson and Webster, one must examine the results to make sure that the constitutional minimum is satisfied that each state receive at least one representative. If not, then assign the seat, modify (increase) the divisor, and go back to Step 3. We caution that for enacting apportionment acts Jefferson s method historically was never used in the context of a modified divisor method but only in the context of a basic divisor method. We illustrate the algorithm by reworking the Delaware example using Jefferson s method and the Rhode Island example using Webster s method. We will always round the standard divisor down for computational purposes within the modified divisor algorithm. Example 1 We apply the Jefferson modified divisor method for Delaware. Step 1. The House size is 41. Step 2. The standard divisor for Delaware = /41 = Step 3. Calculate each state s quotient using the given divisor. Step 4. Round each state s quotient to obtain each state s apportionment. For the Jefferson method all quotients are rounded down to obtain the number of seats. Census 2010 Jefferson's Method County Population Quotient Seats Kent New Castle Sussex Delaware Step 5. If the apportionments all sum to h, then DONE; ELSE, modify the divisor and GO TO Step 3. Jefferson s method with divisor yields a House size of 40, one short of our goal of 41. Hence, since the resulting House size is too small, then our divisor, was too large. We modify the divisor down to and go back to Step 3. Census 2010 Jefferson's Method County Population Quotient Seats Kent New Castle Sussex Delaware Voila! Success! This time, the divisor of produces the desired House size of 41 using Jefferson s method. A key thing to note with the modified divisor method is that we don t have any left-over seats to distribute; hence, there is no need for a priority list. By modifying the divisor within the basic divisor 28

32 method, all the seats are distributed. We merely need to modify the initializing standard divisor to obtain the desired House size goal. Example 2 We apply the Webster modified divisor method for Rhode Island. Step 1. The House size is 75. Step 2. The standard divisor for Rhode Island = /75 = Step 3. Calculate each state s quotient using the given divisor. Step 4. Round each state s quotient to obtain each state s apportionment. For the Webster method all quotients are rounded normally. Census 2010 Webster's Method County Population Quotient Seats Bristol Kent Newport Providence Washington Rhode Island Step 5. If the apportionments all sum to h, then DONE; ELSE, modify the divisor and GO TO Step 3. Webster s method with divisor of yields a House size of 76, one more than our goal of 75. Hence, since the resulting House size is too large, then our divisor, was too small. We modify the divisor up to and go back to Step 3. Census 2010 Webster's Method County Population Quotient Seats Bristol Kent Newport Providence Washington Rhode Island Voila! Success! This time, the divisor of produces the desired House size of 75 using Webster s method. Joseph Hill made his original proposal for adjusting the quota method from Hamilton s major fractions to priority numbers adjusted by a geometric mean. Shortly after Hill presented his idea, Edward Huntington, a mathematician who taught in the Engineering Department at Harvard, adapted Hill s idea to a modified divisor method by presenting another criterion for rounding a decimal. Suppose that q is a positive decimal number with a nonzero decimal fraction. Let n be q rounded down. Then q rounded up is n+1. Huntington s rounding criterion is round q up if and only if q > GM(n,n+1), the geometric mean of the round down, round up options. Accordingly a modified divisor method 29

33 incorporating this method of rounding is called the Huntington-Hill method in today s apportionment literature. To illustrate we rework the Rhode Island Example using the Huntington-Hill method. Example 3 We apply the Huntington-Hill (H-H) modified divisor method for Rhode Island. Step 1. The House size is 75. Step 2. The standard divisor for Rhode Island = /75 = Step 3. Calculate each state s quotient using the given divisor. Step 4. Round each state s quotient to obtain each state s apportionment. For the Webster method all quotients are rounded normally. The GM column is the geometric mean of the round down, round up options for the quotient. If Quotient > GM, then round up; otherwise, down. Note that the results are the same for Huntington-Hill and Webster at this point. Census 2010 H-H Method: County Population Quotient GM Seats Bristol Kent Newport Providence Washington Rhode Island Step 5. If the apportionments all sum to h, then DONE; ELSE, modify the divisor and GO TO Step 3. The H-H method with divisor of yields a House size of 76, one more than our goal of 75. Hence, since the resulting House size is too large, then our divisor, was too small. We modify the divisor up to and go back to Step 3. Census 2010 H-H Method: County Population Quotient GM Seats Bristol Kent Newport Providence Washington Rhode Island Voila! Success! This time, the divisor of produces the desired House size of 75 using the Huntington-Hill method. In 1929 Congress enacted apportionment legislation that froze the House size at 435 for any apportionment based on the decennial census. House size 435 is still in use today. Further, in a 1941 amendment to the 1929 act, Congress permanently adopted the method of Huntington-Hill to apportion 30

34 the U. S. House of Representatives. Computations are automatically done by the U. S. Census Bureau upon receipt of the census. Things will remain this way until Congress changes the law. An obvious inquiry at this point is, how did Congress do at coming up with a workable methodology? Is there a best method for apportionment? The latter question was answered by two mathematicians, Michel Balinski and H. Peyton Young. The Balinski and Young Impossibility Theorem In 1982 Michel Balinski and H. Peyton Young proved that there is no perfect apportionment method. The Balinski and Young Impossibility Theorem There are no perfect apportionment methods. Any divisor method may violate the quota rule. Any method that avoids the Alabama paradox may produce quota rule violations. Further, any quota method is subject to paradoxes, especially the Alabama paradox. The two main apportionment method flaws that Congress discovered in history were quota rule violations and the Alabama paradox. The Balinski-Young Theorem shows that it is impossible to avoid both. Any divisor method, whether basic or modified, may produce quota rule violations. The dealbreaker for the quota method was the Alabama paradox. However, the cost of avoiding this paradox is to use a method that may produce quota rule violations. To absolutely avoid quota rule violations, one must use a quota method, which may produce unwanted paradoxes. The research of Payton and Young has shown that the Webster modified divisor method produces the best results as a general method. The Webster method of rounding is the only neutral quotient rounding mechanism. Adams, Dean, and Huntington-Hill all have an inborn bias that when it appears always favors small states over large states. Jefferson has an inborn bias that when it occurs always favors large states over small. Further, of all divisor methods Webster s is the least likely to violate the quota rule. It is not only least likely, but also unlikely. Webster can expect to produce a quota violation in the decennial reapportionment of the US House of Representatives once in about years. 12 Exercises 1. Repeat the exercises in the previous section, The Quota Method, by applying the modified divisor methods of A. Jefferson, B. Webster, C. Dean, D. Huntington-Hill, E. Adams. 2. In any of the problems you worked in Exercise 1, were there any quota rule violations? 12 Michel Balinski and H. Peyton Young, Fair Representation: Meeting the Ideal of One Man, One Vote, Brookings Institution Press, Washington, D. C., Second Edition, 2001:

35 Section 5: It s a Matter of Priority In this section we study the method currently used for computing congressional apportionment. Then we look at some proposals currently afloat for reforming the current situation. Priority Calculation Techniques In the previous section we studied the modified divisor method for apportionment. The method engaged a specific technique of computation in that apportionment was calculated for a specific House size. During debate based on the 1910 census, Walter Willcox introduced a serial method of computation that avoids the need to rerun the algorithm each time one wants to investigate a different House size. 13 The serial computation technique is one used today. We recommend that the reader pause to view the short (under two minutes) and charming video, The Amazing Apportionment Machine, available on the website of the U. S. Census Bureau. 14 Willcox s technique first assigns one seat to each state. This immediately satisfies the Constitution s minimum requirement that each state shall have at least one seat. In today s House with 435 seats and 50 states, this initial step distributes 50 seats. Then there are 385 more seats to be distributed. Willcox asks, which state merits the 51 st seat? 52 nd seat? 53 rd seat? Etc. To determine the serial merits of seats in the House, Willcox computed a set of priority numbers. The priority number (PN) for a state to receive an n+1 st seat given that a state has n seats is defined by PN(n) = state population ave(nn,nn+1) The various methods for computing an average lead to the various methods of apportionment. Jefferson: ave(n,n+1) = max(n,n+1) Adams: ave(n,n+1) = min(n,n+1) Webster: ave(n,n+1) = AM(n,n+1) H-H: ave(n,n+1) = GM(n,n+1) (H-H means Huntington-Hill) Dean: ave(n,n+1) = HM(n,n+1) For example, to compute the Webster priority number for a state to receive a second seat, divide a state s population by AM(1,2) = 1.5; for a state to receive a third seat (given that it already has two seats), divide the state s population by AM(2,3) = 2.5; etc. The averages needed to obtain the priority numbers for the different apportionment methods are: Jefferson: Adams: Webster: H-H: Dean: 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 1.5, 2.5, 3.5, 4.5, 5.5, 2, 6, 12, 20, 30, 4/3, 12/5, 24/7, 40/9, 60/11, 13 Charles Biles, Congressional Apportionment Based on the Census : Available as an openresource download from or 32